Research On Phaseless Inverse Discrete Hilbert Transform And Compressive Phase Retrieval

In this paper,we study two phase retrieval problems for real-valued signals.The measurement vectors of phaseless inverse discrete Hilbert transform do not satisfy the complement property,a traditional requirement for ensuring the uniqueness of phase retrieval in finite dimensional real-valued space.Therefore,this problem is essentially different from the traditional scheme of phase retrieval.On the other hand,as a special kind of phase retrieval problem,compressive phase retrieval is to deal with sparse signals.A compressive phase retrieval scheme based on a convex optimization called SparsePhaseMax can recover sparse signals with high probability.However,it has a high requirement for the sparse degree of sparse signals.We first address the uniqueness of phaseless inverse discrete Hilbert transform.And then,based on the SparsePhaseMax optimization problem,we study the improvement scheme for the above compressive phase retrieval problem from the three aspects of the application range of sparse signals,the number of measurements and the retrieval probability.Its main contents as follows:Firstly,for the phaseless inverse discrete Hilbert transform related to compactly supported functions,conditions are given to ensure the uniqueness.A condition on the step size T of discrete Hilbert transform is crucial for the insurance.Besides,we address the uniqueness related to non-compactly supported functions.Secondly,the result is on the determination of the signal HΦ in shift-invariant spaces V(HΦ)by phaseless inverse discrete Hilbert transform.Note that measurements is approximative phaseless sampling of HΦ when T>0 is sufficiently small.By establishing a retrieval algorithm,the signals in Hilbert shift-invariant space of the cardinal Bspline of order 3 is used as an example for numerical simulation to examine the efficiency of phaseless inverse discrete Hilbert transform.In the last,we establish a scheme for the choice of the parameter for the SparsePhaseMax,an optimization problem for the compressive phase retrieval.Compared with the existing SparsePhaseMax,we proved that the SparsePhaseMax with the parameter from the proposed choice scheme,holds for more sparse signals.Moreover,the lower bounds of the success probability and the number of measurements are improved.When measurements are relatively few,the improvement of the proposed choice scheme performs better in the simulation.